1. Field of the Invention
The present invention relates to a method for optimizing the development of a heterogeneous porous medium by enhanced recovery of a fluid in place, by determining the position of the front separating a sweeping fluid and the fluid in place.
2. Description of the Prior Art
All the notations used to describe the prior art and the invention are defined at the end of the description.
Describing and simulating multiphase flows in underground reservoirs is fundamental to a reservoir engineers' skill in working with petroleum or gas companies (or, similarly, in water adduction companies). The presence of subsoil heterogeneities and the increasing complexity of drainage systems make a simple analytical solution impossible, which requires development of numerical solutions using a gridded model. The simulation of multiphase flows in a heterogeneous porous medium can then require considerable computing resources, in particular when the numerical model of the medium considered is greatly detailed. This is the case for reservoir engineering in the petroleum field. The cost is mainly due to the computer-based solution of large-size linear systems from the equation that governs the pressure, which has to be updated by following the fluid displacement in order to reach a solution of good precision.
In a two-phase flow in a heterogeneous porous medium, the displacement of a fluid in place (oil for example) is considered under the effect of the injection of another fluid (water for example). The generalized Darcy's law that governs the motion of fluids is written as follows, by means of standard hypotheses and notations:
                                          u            nw                    =                                    -                              λ                nw                                      ⁢                          ∇                              p                nw                                                    ,                                  ⁢                              λ            nw                    =                                    Kk              rnw                                      μ              nw                                                          (        1        )                                                      u            w                    =                                    -                              λ                w                                      ⁢                          ∇                              p                w                                                    ,                                  ⁢                              λ            nw                    =                                    Kk              rw                                      μ              w                                                          (        2        )            u=unw+uw  (3)
where subscripts nw and w designate the nonwetting and wetting fluids respectively. The pressure difference between the two fluids is denoted by pc(S)=pnw−pw. The relation is as follows:Snw+Sw=1  (4)
It is therefore possible to write all the functions depending on saturation S in terms of saturation SW.
By disregarding the capillary pressure, the relationship is:pnw=pw=P  (5)
The total velocity then is written as follows:u=−λ(Sw)∇p  (6)with:∇.u=0  (7)
This system of equations (6) and (7) defines the pressure equation.
In equations (1) and (2), functions kr are the relative permeabilities. The function: kr=kr(Sw). Functions λnw are the respective mobilities of the fluids, and λ(Sw)=λnw(Sw)+λw(Sw) defines the total mobility, which thus explicitly depends on SW.
The mass conservation laws relative to each fluid are written as follows:
                                          ϕ            ⁢                                          ∂                                  S                  nw                                                            ∂                t                                              +                      ∇                          ·                              u                nw                                                    =        0                            (        8        )                                                      ϕ            ⁢                                          ∂                                  S                  w                                                            ∂                t                                              +                      ∇                          ·                              u                w                                                    =        0                            (        9        )            
The system of equations (8) and (9) defines the saturation equation. The term, φ is the porosity (assumed to be uniform in the reservoir), t the time and un, nw is the velocity of the fluid considered.
One of the difficulties in solving these equations comes from the coupling between equations (6) to (9) that couple the saturation to the pressure field. These effects are well known and they control the development of possible viscous instabilities. In particular, in the 1D case, considering sweeping of an initially oil-saturated medium with water, this system of equations can be solved by the method of characteristics, whose solutions are governed by the existence of oscillations corresponding to saturation jumps propagating in the medium.
The discontinuity is governed by a water saturation changing from Swi to Sf, the saturation at the front whose value is given by a Rankine-Hugoniot type condition. In the petroleum context, construction of the Welge tangent allows graphical determination of Sf. Behind this front, a “rarefaction wave” provides, upon injection, transition between Sf and the water saturation upon injection.
There are several known techniques for determining the simulation of multiphase flows in heterogeneous porous media. These methods are all based on the solution of the following system of equations:
  {                                                                                          Pressure                  ⁢                                                                          ⁢                                      equation                    :                                                                                                                        u                  =                                                            -                                              λ                        ⁡                                                  (                                                      S                            w                                                    )                                                                                      ⁢                                          ∇                      p                                                                                                                                (            10            )                                                                                                                                ∇                                          ·                      u                                                        =                  0                                                                                                      Saturation                  ⁢                                                                          ⁢                                      equation                    :                                                                                                                                                  ⁢                          (              11              )                                            ⁢                  ⁢                                                                      ϕ                ⁢                                                      ∂                                          S                      nw                                                                            ∂                    t                                                              +                              ∇                                  ·                                      u                    nw                                                                        =            0                                                (            12            )                                                                                          ϕ                ⁢                                                      ∂                                          S                      w                                                                            ∂                    t                                                              +                              ∇                                  ·                                      u                    w                                                                        =            0                                                (            13            )                              
The techniques differ in the method used for solving this system. Generally, in a complex heterogeneous medium, this type of system of equations has to be solved numerically.
Oil companies most often use conventional discretization techniques of finite volume type (Aziz, Kb., Settary, A.: Petroleum Reservoir Simulation. Applied Science Publishers, London, 1979). Concerning temporal discretization, there are many choices: if numerical robustness is favored, pressure as well as saturation can be totally implied. If oscillations of the solution are preferably avoided and higher accuracy is desired, an implicit scheme will be selected for pressure, and explicit for saturation. The existence of fronts is generally observed which are governed by the same discontinuity of the saturation, whose value suddenly changes from the irreducible water saturation, Swi, to the front saturation Sf. These discontinuities are due to the hyperbolic character of the saturation transport equation and are described analytically in the case of the 1D displacement: it is the Buckley-Leverett theory that is described in detail in Charles-Michel Marie's work Les Écoulements Polyphasiques en Milieu Poreux. Seconde Édition Revue et Augmentéé 1972. Editions Technip, Paris.
Whatever the method which is selected, the pressure equation has to be regularly updated to precisely estimate the velocities.
Another class of techniques, which are more and more commonly used by operator companies, comprises the techniques referred to as “streamline,” with one reference being: R. P. Batycky, M. J. Blunt, and M. R. Thiele, A 3d Field-Scale Streamline-Based Reservoir Simulator, SPERE, 11, 246-254 (1997).
In these methods, the saturations are updated by following the displacement of the fluids in their motion along the streamlines, parametrized curves defined by dx/dt=u(x,t). A variable change allows returning to the aforementioned Buckley-Levereft theory. However, in order to properly calculate velocity u(x,t), the pressure is solved as many times as required for accuracy reasons, on a Cartesian grid. These methods are faster than conventional methods. They do not have the versatility and the robustness for dealing with complex compositional problems where the conventional method remains the only possible solution. Costly pressure field updating cannot be avoided.
Finally, there are front tracking techniques (R Juanes a,d K A Lie “A Front Tracking Method for Efficient Simulation of Miscible Gas Injection.” paper SPE 92298, Houston 31 Jan. to 2 Feb. 2005) are known which monitor the front by a Lagrangian approach requiring the same pressure updating. In this type of method, the description of multiphase flows in underground reservoirs determines the position of the front (also referred to as interface) separating two immiscible fluids in motion which are a fluid in place (oil for example) and a sweeping fluid (water for example), also called injected fluid.
The evolution of the front in the reservoir as it flows therethrough is considerably influenced by coupling (referred to as viscous coupling by reservoir engineers) between the pressure field and the saturation field (Saffman P. G. and G. Taylor. The penetration of a fluid into a porous medium or hele-shaw cell containing a more viscous liquid. Proc. Royal Society of London, A245:312-329, 1958, Nœtinger B., V. Artus, and L. Ricard, Dynamics of the Water-Oil Front for Two-Phase, Immiscible Flows in Heterogeneous Porous Media. 2-Isotropic Media. Transport in porous media, 56:305-328, 2004).
In particular, when the injected fluid is less viscous and consequently more mobile at the front than the fluid in place, the viscous instabilities will always favour flow of the fluids in the most permeable layers of the medium. The breakthrough time through these layers is much faster than in the rest of the reservoir. On the other hand, if the injected fluid is less mobile, the viscous coupling can slow the injected fluid down in the initially higher-velocity layers, thus compensating for the permeability differences due to the stratification. A stationary front then appears (Artus V., B. Nœtinger, and L. Ricard, Dynamics of the Water-Oil Front for Two-Phase, Immiscible Flows in Heterogeneous Porous Media. 1-Stratified Media, Transport in Porous Media, 56:283-303, 2004).
It is well known that the front stability after small perturbations is determined by the fluid viscosity ratio. If the less viscous fluid is displaced by the more viscous fluid, then the front is stable, and vice versa. If the situation is unstable, phenomena referred to as “viscous fingers” develop in the course of time.
The effects of the relative permeability are taken into account by considering the Buckley-Leverett type displacements in a homogeneous medium. Such displacements are described by a hyperbolic saturation equation. One solution to this equation using the method of characteristics provides solutions that have oscillations characterized by a Rankine-Hugoniot condition. It has been shown that the stability of the front where the saturation has a “shock” is determined by a frontal mobility ratio, denoted by Mf:if Mf<1 for which the front is stable in the face of small perturbations, and vice versa. This frontal mobility ratio can be related to the viscosity ratio using the relative permeability function.
More recently, this result was confirmed within the context of linearized stability analyses of the water injection process. In this context, it has been shown that the evolution of the Fourier transform of the front position fluctuations, the latter being denoted by □h(q, t), in a homogeneous medium is given by:
                                                        ∂              t                        ⁢            δ                    ⁢                                          ⁢                      h            ⁡                          (                              q                ,                t                            )                                      =                              c            0                    ⁢                                  q                                ⁢                                                    M                f                            -              1                                                      M                f                            +              1                                ⁢          δ          ⁢                                          ⁢                      h            ⁡                          (                              q                ,                t                            )                                                          (        14        )            where:
      c    0    =                              u          0                ϕ            ⁢                                                  f              w                        ⁢                          (                              S                f                            )                                -                                    f              w                        ⁡                          (                              S                wr                            )                                                            S            f                    -                      S            wr                                =                            u          0                ϕ            ⁢                        f          w          ′                ⁡                  (                      S            f                    )                    with:                Mf is the frontal mobility ratio;        u0 is the mean filtration rate along the direction of axis X;        φ is the porosity;        ƒw is the Buckley-Leverett function representing the fractional water flow;        Sƒ is water saturation at the front; and        Swr is maximum water saturation.In summary, all the aforementioned numerical techniques are costly in computing time, mainly because of the large number of solutions of large-sized linear systems from the equation governing pressure, that have to be updated by monitoring the displacement of the fluids with good precision.        
Existing solutions do therefore not allow obtaining quantitative answers to practical questions that petroleum engineers consider when carrying out reservoir simulation studies, that is:
1. Fast sampling of the space of the study parameters that the modelling relates to, is important to optimize one or more economic and technical development criteria (development decision, selection of the position of the wells, of a recovery method, etc.).
2. Knowledge of how to coherently modify the geologic model is important in order to best calibrate the observed production data, including repeated seismic survey data. This can allow drilling while adapting to the geologic heterogeneities, to locate the fluids and therefore to better control the recovery scenario.
3. The ability to estimate uncertainties by carrying out Monte Carlo simulations is important to test the role of heterogeneities or of little-known parameters characterizing the subsoil. This allows quantification of the risk level linked with the development of a reservoir and thus to redefine development parameters, or even the economic parameters.
An exhaustive study including these various aspects (sensitivity study, data calibration and uncertainty estimation) can then potentially require repeated use of a flow simulator (dedicated software), hence the advantage of fast computing tools, of a precision compatible with that of the input data.
The method according to the invention allows determination of the position of a front separating two immiscible fluids in motion in a heterogeneous porous medium, without systematically updating the pressure field so as to obtain a simulation of the multiphase flows in the porous medium that is fast enough to obtain quantitative information allowing optimum development of the reservoir, by determining technical development parameters and/or economic parameters.